The global wave front set of tempered oscillatory integrals with inhomogeneous phase functions

We study certain families of oscillatory integrals $I_φ(a)$, parametrised by phase functions $φ$ and amplitude functions $a$ globally defined on $\mathbb{R}^d$, which give rise to tempered distributions, avoiding the standard homogeneity requirement on the phase function. The singularities of $I_φ(a)$ are described both from the point of view of the lack of smoothness as well as with respect to the decay at infinity. In particular, the latter will depend on a version of the set of stationary points of $φ$, including elements lying at the boundary of the radial compactification of $\mathbb{R}^d$. As applications, we consider some properties of the two-point function of a free, massive, scalar relativistic field and of classes of global Fourier integral operators on $\mathbb{R}^d$, with the latter defined in terms of kernels of the form $I_φ(a)$.

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